The Behajaina method applied to the bodies of fractions twisted by a derivation
type: Item
instance of: scholarly article
title: The Behajaina method applied to the bodies of fractions twisted by a derivation
zbMATH Open document ID: 1482.12002
author: Bruno Deschamps
published in: Research in Number Theory
publication date: 2021-06-11T00:00:00Z
review text: The Inverse Galois Problem (IGP) over a commutative field \(k\) asks whether every finite group is the Galois group of a Galois extension of \(k\). Going back to Hilbert and Noether, the original problem over \(\mathbb{Q}\) remains widely open. Traditionally, this problem has been investigated only over commutative fields. However, there is a notion of Galois extension for arbitrary extensions of skew fields, which is due to Artin: an extension of skew fields \(L/K\) is \textit{Galois} if \(K\) is the invariant subfield of \(L\) under the natural action of \(\Aut(L/K)\). Moreover, when the degree of \(L/K\) is finite, there is also a Galois correspondence generalizing the classical one [\textit{P. M. Cohn}, Skew fields. Theory of general division rings. Cambridge: Cambridge Univ. Press (1995; Zbl 0840.16001)]. Hence, it makes sense to study the IGP over any skew field. \textit{B. Deschamps} and \textit{F. Legrand} [J. Pure Appl. Algebra 224, No. 5, Article ID 106240, 13 p. (2020; Zbl 1480.12004)] started the investigation of this problem and proved that: if \(H\) is a skew field of finite dimension over its center \(k\), which contains a so called ``ample field'', then the IGP has a positive answer over the skew field of rational fractions \(H(t)\). The proof of their result is based on scalar extensions method by using reduced norms. \textit{A. Behajaina} [J. Pure Appl. Algebra 225, No. 4, Article ID 106549, 10 p. (2021; Zbl 1473.12002)] generalized this result to skew fields of rational fractions \(H(t,\sigma)\) (here \(th=\sigma(h)t\) for all \(h \in H\)), where \(\sigma\) is an automorphism of \(H\) of finite order such that the invariant subfield \(k^{\sigma}\) of \(k\) contains an ample field. In this result, the condition that \(H\) is of finite dimension can be relaxed. The key ingredient of the proof is a certain ''embedding'' into the skew field of series \(H((t,\sigma))\), which replaces the use of ''reduced norms'' by Deschamps and Legrand, which heavily…
reviewed by: Angelot Behajaina
Mathematics Subject Classification ID: 12E15
Mathematics Subject Classification ID: 12F12
zbMATH DE Number: 7357694
zbMATH Keywords: inverse Galois problem
zbMATH Keywords: skew fields
MaRDI profile type: Publication
full work available at URL: s40993-021-00263-z
OpenAlex ID: W3164724951
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DOI: 10.1007/S40993-021-00263-Z
description: scientific article; zbMATH DE number 7357694